Prof. Nathan Aviezer wrote a book "בראשית ברא" answering questions about the creation in torah view.
There he claims the 6 days of creation to be 6 periods of time that didn't necessarily last 24 hours. Moreover - maybe it was one cycle of dark&light, but things happened at a quicker pace.
Take, for example, a person's growth. If you were to calculate the rate of growth during his first two years, you would conclude that at the age of 20 he'd be as tall as a skyscraper. Similarly, scientists calculate age of things according to he worlds' pace today, but things may have worked differently in the past.
Hence, dinosaurs might have lived and died prior to Adam Harishon. Maybe created on "day" 5 and extinct before 6.
(Somewhat related: What is the meaning of יוֹם (yowm) in Bereshit?)
The Malbim (on Noach 7:23) addresses "large animals" that couldn't be "wiped out" by the flood, but were buried in depths of the earth following quakes the Mabul created. He is clearly addressing dinosaurs since he's talking about (my free translation): "... geologists who dig and find large animals who have become extinct, they use this to show earth existed long before Bereshis creation..."
Some attribute "התנינים הגדולים" (Bereshis 1:21) to the dinosaurs as a reference of their existing (not mentioned in Malbim).
Another opinion I've heard from reliable people (suggested by the Lubavitcher Rebbe in this letter) is that the world was created "old", such as Adam Harishon was created as a man of 20 and not a newborn baby. Hence, dinosaurs could have either existed - or their remains and footprints were melded into the creation.