I have a Mk3 MX-5 and I use the Saris Bones rack. Works a treat and will also fit pretty much any other car you can think of. I have the 3-bike rack, although I've never carried more than one bike on the MX-5.
The comment about the boot lid has some truth - it is a little flexy, so just be careful how tight you pull the straps on the rack. You don't need to have the straps "solid wood" tight, just firm enough that the rack is stable when you give it a little push/pull to check. Also, best to stop after a few minutes and re-check that nothing has loosened off. I would also recommend using some small cloth patches to put under the rubber feet to protect the boot lid paintwork.
I've used it probably between 10 and 20 times or so without any problems, on twisty country road trips and on several long (3 to 4 hr) motorway journeys. It also folds up and fits into the MX-5 boot (well just, with a little persuasion!) so you don't need to leave it outside the car.