Is there any way to attractively paint cabinets that are not solid wood? I cannot afford to replace them at this time.

what are the cabinets made of? Most material can be painted. Ideally you should use a spray gun. Rent at the local rental center. Be sure to clean the cabinets real good with a degreaser BEFORE you sand. The amount of grease on doors and frames can be a lot even though they look clean. Once you sand without cleaning you will sand this stuff right into the wood and the paint will not properly adhere.
Be sure to mask everyting in the kitchen. Spray dust gets everywhere. Without doing so everything will end up the same color.
Be sure to remove all hardware, and doors, drawers. Replace the hardware if you can. New hinges and handles make the job look great.
Use a good quality paint. High gloss works best, but color choice can be anything you like. Good luck and have fun doing it. Its really not that hard, just time consuming.

Thank you so much for your response. I'm not really sure what they are made of - maybe mdf/particle board. They are relatively nice looking, but definitely need updating. Thank you also for the reminder to tape everything.