A couple thoughts on availability and routing, spurred by the thought that March is awfully close to Winter at some altitudes. There's a lot of good news after research, but its worth spending some time being aware.
After some checking, the good news is that except for GCNP-North Rim, the other parks, Zion (but could be icy), Bryce, and the South Rim, (see tab from the north rim listing) are open year-round, barring any recent snows not fully cleared.
Note of caution however, everything I can see (1, 2) says Cottonwood Canyon Road is a dirt road, and recommended for 4-wheel drives during the winter. Car rental policy is likely to prohibit driving on such a surface.
An alternative is to stay on US-89 after getting into Utah from the GC. About 60+ miles West is Kanab, where you can inquire about road conditions to Zion and Bryce. Although Zion is closer (via Utah-9) at that point, parts of it may be least accessble. I recall in 2003 it was closed until sometime in May because of unusually heavy snow the prior winter. Zion has two sections, not connected within the park itself -- that is you can drive through one section for some amazing visual memories from the road. The other section has a visitors center and a number of trails/tracks. Still you should be able to biew at least some of it.
Bryce is further North via US-89 and Utah-12 towards Bryce. If you have the time, you can go further East on Utah-12 from Bryce and touch the Northern reaches of Grand Staircase-Escelante then doubling back on Old-Escelante Road. If you want a different route back, you could try US-89 N, Utah-20, and I-15 back down to Utah-9 (for Zion) or continue on to L.V.
In all cases its worth checking locally at time of travel to be sure of the road conditions, and of course defer to anyone who's travelled in these parks during that time of year.
A few other notes:
Watch your fuel level. While most parks have a village nearby with filling stations, there are no fuel facilities for cars within the parks .
Daylight Savings Time begins March 11, in 2007, due to recent national legislation. A timezone runs roughly along the Eastern Nevada and California borders. Arizona doesn't participate, so after March 11, when the most Western states "spring forward" Arizona's time will be synched with theirs. Utah, however does particpate, always remaining an hour ahead of them.