"Golden Hour" is great for a number of different reasons beyond the dramatic. It gives the light direction in a manner that doesn't cause things like a severe shading of eyes and such.
Me, I prefer to shoot landscapes in the mornings. Get up at 5, out the door by 5:20. Enough time to set up the camera where I'll be shooting and kick back and drink coffee, all by myself. When you have a 3 year old, the latter is really the most important part.
I could never get up that early in the morning, so I'd go for sunset rather than sunrise. Generally, however, I prefer overcast days - controlling exposure seems so much easier then, and you don't really have to worry about unwanted shadows cast somewhere into your beautiful composition; so I'm following a somewhat practical approach here.
I'll shoot throughout the day but I prefer civil twilight before sunrise and maybe about a half hour after. About 30 minutes before sunset and through civil twilight afterward. The colors are soft, rich and show well with long exposures.
Overcast days are good for you folks with SLRs, but really stink for me. ISO 400 stinks and 800 is completely useless and I don't like to carry a tripod everywhere I go (mainly because my photography outings are also kid outings to the beach or zoo or something). 1/40 shutter with 380mm (35mm equiv) on overcast days just doesn't cut it. I need light and lots of it.
Nice days with some clouds are fine when the wispy clouds move past the sun, but complete overcast means no shooting for me.
I just shot this past weekend. Super bright, not a cloud in the southern california sky. I shot throughout the day in a public park from 9am-sunset. I found that shooting shooting about 1.5 hrs before and after noon was a bit tricky to handle. An assistant with a reflector helped. For the rest of the day, I spent mostly shooting with a fill-flash when needed. It worked out quite nicely.
*note* if you are the reflector assistant, be careful... you have a high chance of getting sunburnt. I learned this the hard way >.<