Three men allegedly tried to get into the flats by telling a ground floor resident they had come to play football with a friend in the building, but were denied entrance.
The suspects, two were said to have Irish accents, eventually managed to get into one of the flats, but left after being disturbed by a neighbour.
A laptop, computer and stereo were among several items stolen.
One of the suspects was aged about 16, with short dark hair and was wearing a navy blue waxed jacket. The second man was aged about 45 and was wearing a blue denim shirt.
The third was also a white man, though no other descriptions were available from police.
Police are asking anyone with any information about the thefts to call Kingston CID on 020 8247 5913 or Crimestoppers anonymously on 0800 555 111.