A fair question.
They didn't accept it at face value. I was (and am--I spent today leading a sampling effort, in fact) require to extensively document my work, in a bound field notebook in indellible ink (black or blue). I submitted the samples under an unbroken chain of custody(COC). There were also others who were also documenting the proceedings, including a photo survey of the trenches we took the samples from. They're not exactly accepting my word at face value; technically, they're accepting the COC (in fact, that's legally what they're doing).
If it's proven that my samples--ANY samples, from the first ones I ever touched to the ones I shipped before coming home this evening--were fraudulant, my career is OVER and I'm probably going to spend time in jail. The mere accusation of inaccuracy can be enough to get me fired. There are EXTREME consequences to this sort of thing, and field geologists take equivalently extreme care with our sampling. The samples I took today--standard dirt samples--were documented in four places, with copies sent to several people. And this is minor stuff, just fact-checking.
Also, as IanS said, I'm not exactly working with world-famous relics here. I was sampling bits of charcoal (almost certainly natural) in a trench in the middle of nowhere. Those C14 samples were minor samples, and required minor documentation. The more serious the sample, the more documentation required. Sampling a world-famous relic is a pretty major sample. Furthermore, I've established myself as someone capable of taking these samples, and competant at standard sampling procedures (EPA Region 9 and standard paleontological sampling, to be specific), via training under recognized experts. And again, there's the chains of custody--they are unbroken, and included in the reports that discuss the lab results (one appendix is always the lab results, and within that is the COC for those samples).
THAT is the major issue: the chain of custody. My samples are valid because, in major part, that chain is documented and unbroken. The chain of custody for the sticky-tape fibers, however, is not--meaning that no one can prove the fibers tested were actually fibers from the shroud. We could, if they were available, do trace-chemical analysis on them to prove it, but this has not been done to my knowledge, so the chain remains broken. The samples, simply put, are inadmissible because we don't know where they're from.
If Rogers produces an unbroken chain of custody, this objection disapears. Until then, we simply cannot know where the samples came from. He SAYS they're from the shroud, but in science it's not enough to say something. You need to prove it.
I want to emphasize that I'm merely giving this to provide the information; it's been an agravating day, so my tone is probably not the best. Please don't read any anger into this--I'm happy to answer the question.