You can't get rid of them easily in post processing... you can lighten them, but not really get rid of them.
The key is to take the image paying attention to the shadows and maybe even use them to enhance the image. You didn't say of you are talking about snapshots or more formal photos...
Pros will use various diffusers and reflectors to adjust the light. Depending on your camera, you may be able to "force" the flash to fill in dark areas. If you shoot with a DSLR, learn how to expose manually. "Automatic" settings suck in weird lighting.
You point out the main reason we prefer to shoot early and late in the day... Harsh light and the resulting shadows and glare are not photographers' friends.
I should add that 99% of my pro work is underwater, but oddly, shadows are an issue there as well. I always use two very strong flashes to fill in the darker bits, the key being to be able to balance the artificial light with natural. I often shoot directly into the sun underwater (where it's much less intense of course) but this leaves foreground subjects in the dark (so to speak). This is where the flashes come in...