Streaming codes offer reliable recovery under decoding-delay constraint τ, of packets transmitted over a burst- and-random-erasure channel. Prior rate-optimal code constructions had field size quadratic in τ and employed diagonal embedding of a scalar block code of length n within the packet stream. It is shown here that staggered diagonal embedding (SDE) under which the n code symbols are dispersed across a span of N ≥ n successive packets leads to a simpler, low-complexity construction of rate-optimal streaming codes having linear field size. The limits of the SDE approach under the restriction N ≤ τ + 1 are explored. Some binary streaming codes that are rate-optimal under this restriction are identified.