Coodem,
Luton shouldn't be a problem - I regularly go through, and like you am based at Cranfield.
Best advice I can give:
1) As soon as possible after departing Cranfield, switch to Luton - it gives them plenty of time to deal with you before you're hitting their zone
2) As others have said - sound confident - it instills confidence that you will listen and follow their instructions correctly!
3) Again, as others have said, expect to cross the threshold of the runway in use. Luton's runway is almost the same as Cranfield's, so if you depart Cranfield on 21, expect to cross the Eastern threshold of Luton, and vice-versa
4) Hold your altitude well, and follow all their instructions.
5) If you can, try and avoid their busy times - you have much more chance of a transit when they don't have millions of Easy's / Ryan's inbound
6) Enjoy it - it's really cool flying over and looking down at all the Easy's etc!
I've never once been refused a transit - occassionally held for a couple of minutes, but always accommodated by friendly controllers.
I know this all sounds stupid to most, but I just want to make sure I’m prepared.
Nothing stupid at all atbout seeking advice to improve your knowledge.