Edinburgh are through to the last four after stunning four-time champions Toulouse with a 19-14 win at Murrayfield but Bradley insists 1999 winners Ulster must be favourites to book a place in the Twickenham Stadium final on Saturday, May 19.
"Ulster have beaten us twice in the domestic league this season and I guess that makes them favourites," said Bradley, still in his first season with the Scottish club.
"And we can accept that because they have proved they deserve to be in that position going into the semi-final.
"Playing at home is another factor in their favour and we will have to deal with that on the day - and there are different ways of doing that - but we are used to going into matches as underdogs so I don't think it will be a big issue for us.
"As to whether we would prefer to be playing a team from outside our own domestic tournament is a tough one and at this stage we are just delighted to be in the semi-final - that is point number one.
"Point number two is that playing against Ulster at Irish rugby's national ground is sure to be a very special occasion. All the other semi-finalists, Leinster, Clermont and Ulster, are three very good sides in their own rights and we are happy enough to be playing Ulster.
"As to whether I am surprised by our success in the Heineken Cup this season all I can say is that we play to our strengths and we have a lot of talented players.
"Against Toulouse in the quarter-finals we had 11 current internationals starting that match and Toulouse had three players who played for France against Scotland a couple of weeks earlier in the Six Nations and you have to look at the facts of the match on the day.
"We know that if we play close to our full potential at home we can trouble anyone and in front of almost 38,000 fans that is how the game unfolded.
"The key strength for Edinburgh Rugby is that, from 1 to 15, they can all play football and that has been the standout feature for us.
"If you go back to the two matches against both Racing Metro 92 and London Irish, to beat both those sides twice we needed to play good football and that is what we achieved in all those games.
"As for Ulster, from 1 to 15 they are very efficient in what they do and very effective in how they do it - and quite a physical side. The intensity they brought to the quarter-final against Munster was a special performance and they are capable of producing very big results.
"In the Pool stages they destroyed Leicester and also turned over Clermont at Ravenhill and they did that with a fair degree of panache, along with the quality of player they have.
"Naturally I am delighted and very proud of what Edinburgh have achieved so far but we have to be pretty calculating on our assessment of what potentially lies ahead.
"We have a great opportunity here. If we play like we did against Toulouse and close to our best we will trouble them and it will be up to them to deal with us. We are looking ahead with a lot of confidence and a fair degree of excitement as well."