We present a general procedure to convert schemes which are based on staggered spatial grids into non-staggered schemes. This procedure is then utilized to construct a new family of non-staggered, central schemes for hyperbolic conservation laws, by converting the family of staggered central schemes recently introduced in [nessyahu-tadmor], [liu-tadmor], [jiang-tadmor]. These new non-staggered central schemes retain the desirable properties of simplicity and high-resolution, and in particular, they yield Riemann-solver-free recipes which avoid dimensional splitting. Most importantly, the new central schemes avoid staggered grids and hence are simpler to implement in frameworks which involve complex geometries and boundary conditions.