  For an arbitrary ideal $I$ in a polynomial ring $R$ we define the notion of
initially regular sequences on $R/I$. These sequences share properties with
regular sequences. In particular, the length of an initially regular sequence
provides a lower bound for the depth of $R/I$. Using combinatorial information
from the initial ideal of $I$ we construct sequences of linear polynomials that
form initially regular sequences on $R/I$. We identify situations where
initially regular sequences are also regular sequences, and we show that our
results can be combined with polarization to improve known depth bounds for
general monomial ideals.
