If your playing a barre chords, and your constantly playing them within a progression. You will need to learn how to relax your hand between switching and when there is no need to tense your hand up. Learning the shape of the chord is one thing, but you also have to learn how much tension you actual need to apply.

Heres a method i use to determine if i am pressing to hard on the strings.

Rest your finger on the string so it is muted and pluck it constantly.
Slowly apply pressure until you hear the note ring out clearly.

Once you have figured out how much pressure is necessary to have the note ring clearly you will end up seeing that it really doesnt require that much at all. If it still seems like you have to press really hard. There could be 2 possible problems that i can think of.

1. Like Allen said, your hand just isnt quite strong enough yet.
2. Your frets are really low.

String gauge can and will affect how much pressure you need to apply to have it ring clearly, but if you use the method above. You will be able to tell how much is actually needed to make it ring nice. :)

Hope this helps.