Sunday's 3-1 defeat to Everton at Turf Moor means that the Premier League newcomers have failed to win any of their first nine matches in the division.
Following the defeat, the Clarets players walked off the pitch to a warm reception from the home supporters.
"I thought the fans were terrific," Dyche told the Evening Express. "I think they get the challenge. With all this coverage now, every fan out there can see it right in front of their face.
"Our club get promoted, and they don't come in thinking, 'Oh well, that's it, we'll just roll up the table'. They look at all the signings, all the teams they're playing against, the quality, the worldwide class that's out there.
"The fans are backing the players and I've told them that's important. I think they've been absolutely fantastic and we will still need them. They wanted it, we wanted it, the town wanted it, it's here, it's real. That didn't mean it was going to be an easy ride."
Burnley remain 19th in the table following the defeat.