You'd need at least one person playing to a click to make this work.
Actually, playing along to pre-recorded material or sequencer parts takes a bit of organising. The band members need to hear the parts clearly in order not to get lost (not that easy without good monitors) and if you deviate at all from the arrangement it'll be one mighty car crash.
Apart from logistics, I personally think it would be the kiss of death for any band trying to sound 'sixties'. 

Lots of cover bands do this, but as Chrisso says, it means playing along to a click or a drum-machine sequence. So at the very least the drummer will need headphones or a good monitor, and can lead the rest of the band with his or her playing.

I've done tons of gigs like these over the years, and continue to do them - they used to bug me, but now I don't mind them. Not every song will lend itself to this, and it can be a challenge to keep the sequence from sounding mechanical. But it can be done very effectively - it's just a VERY different way to play, which you may or may not find enjoyable. And of course there is ZERO margin for error.

I saw a band not long ago that played to some sequenced keyboard parts. They were really tight and even though I didn't like the style, I had a lot of respect for them pulling it off. My guitar player wants to try recording bass parts to his looper, and stomping it during our shows since our bass player just left town for a couple months. I don't really like the idea. Every time I have tried to play along to loops he's made, there's always some point that doesn't stay in time and it throws me off. He wants me to just get used to the out of sync part, but I'd rather just keep it live. If I triggered any loops it would be purely ambient soundscapes.

That's my experience too. I was in a band where the bandleader pre-recorded rhythm tracks/backing vocals for EVERY song, so I had to play listening to headphones for every song. It was tedious and joyless for me. I had to chart out the songs I didn't know exactly because you CAN'T screw up when you're playing to a track. Plus you can't really even hear what you're playing. It sucked all the fun out of playing for me.

Scott -

The way to actually attempt this with the Garageband to iPod set-up you described - and I subcribe to all the "you may not really enjoy it nearly as much as you think you will" warnings - would be to record the oft-suggested click as a part in Garageband along with the other parts.  Then mix it down with all the overdubs hard left and the click hard right - using a splitter run the left side to the PA and the right side to some kind of headphone amp (Rolls makes one; as do others).

The click will let, at least, you stay with the ipod - and will also many the keyboard parts wouldn't have to be continuous.

Be sure to put four extra clicks in front of whatever count-off you'll be using.  For instance, you listen to the first four clicks, then count the band in over the next four, then the song starts (and maybe the extra key parts don't start until the chorus).

It can sound really, but it can also turn the drum chair into the "stay with the click guy", instead of "lead the band guy" - which can certainly be a loss - or at least, just very different.

David