One of the six panels from the 'elaborate but fragile' late 14th century/early 15th century alabaster piece in St Peter's Church, Drayton, near Abingdon in Oxfordshire, was taken between 9.30am on Thursday, March 8, and 11am on Saturday, March 10.
Depicting The Annunciation, it was described as "one of the finest of existing alabaster reredos", by Natalie Merry, secretary to the Diocesan Advisory Committee.
"It was taken in the daytime, we believe, chiselled out of the wall," she added.
"The parish is understandably very upset about it because to them it is obviously more than just a work of art, it is priceless. It is very unusual - these are panels of a larger reredos, which we think would have been behind the high altar showing various elements of the life of Christ, and there would have been more of them that survived.
"These parts were found in the churchyard, when a vault was being dug, and they were reinstalled in the lady chapel, in that current form, which doesn't bear much relation to the historic form."
Ms Merry thinks the panel "may have been stolen to order, or with a specific buyer in mind, given the peculiarity of the piece", but stressed that it was too early to say for sure.
Thames Valley Police believe the carving is worth about £5000, and say anyone with information should call PC Steve Williams from Abingdon police station via the 24-hour enquiry line on 101, or contact the independent charity Crimestoppers anonymously on 0800 555 111.