All good suggestions: Slight start delay between the tracks, EQ settings, vocal technique, and so on. Also chorus and reverb are key things to work with.
Try EQing say 3 different background "Copies". EQ 2 parts fairly close (to each other, but inbetween the 3rd part and the lead vocal), and pan them left and right (doesn't have to be hard panned, but sweep them around to find the right balance, which will be easier towards the end of what I am getting at), and EQ the 3rd to contrast the main vocal and leave it panned close to center, and apply to it a med rate/depth chorus. Also apply chorus to the 2 other parts, but slower (I generally find the further from center you go, the slower you want your choruses. It makes the sound wobbly otherwise, and may create unnescessary phase problems in the mix). As far as delaying the parts you can do that, by shifting samples or a delay plugin. I actually prefer to sing the individual parts and sing the delay, but you can also use a good reverb to add delay. The background parts usually require a longer reverb time then the lead, especially if you are trying to create the illusion of ensemble. A really dry vocal will deemphasize the difference with the background if those parts are fairly saturated (Really this part is a matter of judgement to what vibe you want).
This whole technique works best if you have many different vocal takes to work with.
One other thing to try, is to apply a surround panner to each part, in combination with different panning laws. Even if it is just a stereo part, using this method often will make it easier to get parts to fit because the volume of the sample will be adjusted as it moves through the stereo (Surround) fields, and you can virtually place your singers, with minimal level adjustments (However if using surround panners in stereo, you need to invert the panning motions. Any forward pans backwards, and any backwards pans forward, due the fact that you are trying to feed a signal to speakers that aren't there. So basically if your looking at a surround panner on screen, the bottom would represent the back row of your choir, but left is still left, and right is still right. Unless you have independently adjustable panning laws, but most programs don't).