You can style them, but you can't remove the elements that are already there. If you're creative, you can work with that and do something like this:
input[type=file] {
-webkit-appearance: none;
-moz-appearance: none;
appearance: none;
background: #EEE;
background: linear-gradient(to top, #FFF, #DDD);
border: thin solid rgba(0,0,0, .5);
border-radius: .25em;
box-shadow: inset .25em .25em .25em rgba(255,255,255, .5), inset -.1em -.1em .1em rgba(0,0,0, 0.1);
cursor: text;
padding: .25em;
}
http://jsfiddle.net/zr1x1m2b/1/
I suggest you play around with this code, remove lines, add your own, do whatever until you get something that looks how you like!