United are less than 20 days away from their first Championship fixture at home to Derby County on August 7, and Grayson said: "You could say it's only a pre-season game but it isn't. It's preparation for the start of the season and this is bit of a body blow for us.
"Of course it hurts and it should hurt everyone, mainly the players. A lot of things went wrong and they need to be put right before the season starts.
"This is a wake-up call for one or two players, a major kick-up the backside and a reminder that they are still competing for jerseys. We'll take the disappointment on the chin but I don't think anyone comes out this with much credit."
More than 1,700 Leeds supporters helped to make up a crowd of 2,769 at Gigg Lane, but United's defeat appeared to be inevitable once Bury established a 2-0 lead with an hour gone.
Grayson added: "The fans expect a lot more and, standing here as their manager, I expect a lot more from them.
"If we replicate that performance and attitude at the beginning of the season then no disrespect to Bury but we'll get done in the Championship by seven or eight. We're going to have to learn quickly."