First of all the Santa Ana River, isn't a river. Like most urban rivers in Southern California, its essentially a storm drain, and is dry most of the year except at the headwaters and tidal areas near the "river" mouth, which are also the most scenic areas.

For the most part it is less than scenic and often downright ugly, but you can get a lot of miles in. A lot of Gang Activity in Santa Ana and Anaheim (but they don't seem to Bother Cyclists) and there are occasional homeless camps that appear south of Adams Street in Huntington Beach.

You might want to start your ride early in the morning in the Yorba Linda Area and head toward the beach, before the onshore breeze kicks in (around 11 am there is a nasty headwind if you are riding toward the beach, but it makes for a great ride back)

There is a nice ride than you can pick up at the water tower in Seal Beach and Ride the the Wedge in Newport Beach. Its roughly 30 miles round trip along the beach.
If you decide to ride on PCH, There is construction in the Bolsa Chica Area which willl detour you onto a MU trail (which tends to get crowded with very clueless pedestrians Thursday Evening - Sunday Evening) You can ride the trail the entire way, but life guards will tell you to slow down if they thing you are creating a hazard.

If you want a scenic ride that is relatively hassle free, Ride around Newport's Backbay. It is about 10 mile round trip ride with some nice hills. If you want, connect to Irvine's San Diego Creek Trail which is usually clear of heavy traffic except Saturday and Sunday Mornings. Back and Forth will give you almost twenty miles.