Replacing the logic board is essentially like getting a whole new Mac; though all the data on the hard drive is identical, the new logic board will have different hardware identifiers (specifically, the MAC address) that will tell your Time Capsule, "This is a new Mac that's never been backed up before. Please treat it as such." The Time Capsule, doing as it's told, will fumble along and create a new Time Machine backup while ignoring the old backups completely.
Your options then are these:
1. Scrap your old Time Machine backups and start fresh. There may be an allure to this, but it's almost certainly unnecessary, and you can lose months of perfectly good backups. Plus, you then have to deal with the incredibly long first Time Machine backup all over again.
2. Hack your Time Machine backup using the following procedure, which will allow you to resume Time Machine backups as though your logic board was never replaced.
First, some disclaimers.
Disclaimer 1: If the Terminal's black text on white background fills you with a sense of foreboding and panic, you probably should not attempt this procedure. Typing the wrong thing in the Terminal can result in very, very bad things happening to your computer, from which recovery may be all but impossible. This is the kind of procedure that shouldn't be attempted if you're even slightly uncomfortable doing so. Proceed at your own risk. (cue spooky music and jazz hands)
Disclaimer 2: I didn't come up with this procedure. The extent of my Terminal knowledge can best be described as, "Knows correct answer when told." This procedure was developed by macosxhints.com user kohlmannj, and is reprinted below. It's particularly geared toward Time Capsule owners, but it should work for other types of Time Machine backups as well, with a little modification.
Disclaimer 3: Given #2, we can't fix your machine if you break it. TUAW isn't responsible for user error, wacky non-standard setups that go wrong when introducing a non-native species, etc. In other words, and as with all hacks, you do this at your own risk. (again, cue spooky music and jazz hands)
In the code that follows, everything that's in bold needs to be replaced with values from your own Mac and Time Capsule. "Time Capsule" is the name of the internal disk of your Time Capsule, and "Backup of MyMac" is the name of the sparsebundle disk image volume that mounts when you normally back up to your Time Capsule. You'll want to replace these placeholder values with whatever your volumes are actually named; if your Time Capsule is named "Ren", then replace "Time Capsule" with "Ren"; if your Mac is named "Stimpy", then replace "Backup of MyMac" with "Backup of Stimpy".
The old MAC address (from the old logic board) used in the code below is 00:f9:e8:d7:c6:b5, with the new MAC address (from the new logic board) being 00:1a:2b:3c:4f:56. Note that your MAC addresses will be different; again, the MAC addresses used here are only placeholders. You will need to determine your specific MAC addresses before beginning.
If you don't know how to find your computer's MAC address, then in all honesty, you probably should just stop right now and not attempt this procedure. But if you're really anxious to tickle the Devil's tail anyway, then here's how to find your MAC address:
Go to System Preferences, and select "Network". On the pane that follows, click "Advanced". The first tab on the left should be "Airport" - click it, and your Airport ID will display toward the bottom of the pane. This is the MAC address of the new logic board.
The old MAC address from your old and busted logic board is easier to find - it's the string of numbers and letters after the underscore in the name of the sparsebundle disk image of your old Time Machine backups.
Once you have all the relevant info and have Terminal up and ready, it's time for some UNIXy goodness.
1. Double-click your old sparsebundle disk (MyMac_00f9e8d7c6b5.sparsebundle) image and mount that volume on your Desktop.
2. Input the following commands in Terminal:
$ sudo fsaclctl -p/Volumes/Backup\ of\ MyMac -d
$ sudo xattr -w com.apple.backupd.BackupMachineAddress 00:1a:2b:3c:4f:56 /Volumes/Backup\ of\ MyMac/Backups.backupdb/MyMac
$ sudo fsaclctl -p/Volumes/Backup\ of\ MyMac -e
$ cd /Volumes/Time\ Capsule
$ sudo mv .00f9e8d7c6b5 .001a2b3c4f56
$ sudo mv MyMac_00f9e8d7c6b5.sparsebundle MyMac_.001a2b3c4f56.sparsebundle
3. Quit out of Terminal, eject the Time Machine backup image volume, and manually initiate a Time Machine backup whenever you can leave your Mac in one place and awake for a few hours.
If you're curious what all of the above terminal gobbledygook means, here is macosxhints.com user kohlmannj's translation:
"What we've done here is we've disabled and re-enabled ACLs within the sparsebundle volume (where Backups.db actually is) and used xattr to change the MAC address in the manner shown in the original hint. Next we went to the Time Capsule's volume and changed the MAC address file and the name of our backup to both reflect our new MAC address. I'm fairly certain this should work for Time Capsule-based TM backups...it has worked for me, so best of luck."
Translation of the translation: it tells the Time Capsule to forget about your old, dead logic board and tells it to play nice with the new one. End result: you get to keep all your old backups, and Time Machine will work as though nothing ever happened -- almost.
The first Time Machine backup will see your old backup's disk image mount successfully, but the first backup will seem to take forever, with the status stuck on "Preparing..." for as much as several hours. The reason: since your logic board just got replaced, it's probably been several days since your last Time Machine backup, so Time Machine has to do something called "deep traversal" (which you can see if you fire up Console during the backup) - essentially, your Mac and the Time Capsule have to do a bit-by-bit comparison of pretty much your entire hard drive to find out what's changed since the last backup. Depending on the size of your main drive, this can take a long time -- it took a couple of hours on my wife's MacBook's 120 GB hard drive -- but it won't take nearly as long as starting your Time Machine backups over from scratch would have.