The best information I've seen, and the info I believe, is that John Cheney lived in the Mistley, Manningtree, Lawton region of Essexshire, England, in 1626-1634. This was confirmed by Mr. Leslie Mahler in two articles, one in 2000 in "The American Genealogist" and one in 2008 in "New England Historical and Genealogical Register". (pulling those dates up cold, as my copies are at home; I'm at work). I have also seen a self-published monograph that suggests the John Cheney of Mistley-Lawton was born around 1605 in Waltham Abbey, also in Essexshire, but I haven't tried to confirm that.
John Cheney had two wifes: Amy UNKNOWN, who must have died 1628-1629; and Martha SMITHE mar. 1630 in Lawton. Some web-published pedigrees show John Cheney of Newbury as having married Martha PARRAT in Roxburg, Scotland. This is wrong, I believe, and an attempt to make John's first three children look legitimate (since no record of his marriage to Amy UNKNOWN had been found and thus no one knew of the prior marriage). The first marriage is assumed based on baptism records of Mary, Martha, and John Cheney at Mistley with parents John and Amy Cheney. Mistley is next door to Lawford/Manningtree.
One researcher found the marriage of a John Cheney to Anne LEWELLAN in the Waltham Abbey region around 1624, so if Anne is Amy, that's some further evidence.