Should Christians today fast?

Yes, certainly. The New Testament has its own examples of fasting. Nothing has changed to diminish the value of fasting. It shows God a greater degree of earnestness on your part, and He does not miss the extra effort to seek His blessings. In fact, there are certain situations that cannot be remedied without fasting (Matt 17:21). The Roman Catholic concept of fasting for Lent is hardly fasting by any definition, and it is so steeped in paganism as to be void of value.

See Also

For much detail and encouragement about Bible fasting: Fasting.
For analysis of the need for fasting for a lunatic: Faith Like A Mustard Seed. 
For a Biblical refutation of Ash Wednesday: Ash Wednesday.