We present a model of one-dimensional asymmetric random walks. Random walkers alternate between flights (steps of constant velocity) and sticking (pauses). The sticking time probability distribution function (PDF) decays as P(t) ~ t^{-u}. Previous work considered the case of a flight PDF decaying as P(t) ~ t^{-v} [Weeks et al., Physica D 97, 291 (1996)]; leftward and rightward flights occurred with differing probabilities and velocities. In addition to these asymmertries, the present strongly asymmetric model uses distinct flight PDFs for leftward and rightward flights: P_L(t) ~ t^{-v} and P_R(t) ~ t^{-w}, with v not equal to w. We calculate the dependence of the variance exponent z (sigma^2 ~ t^z) on the PDF exponents u, v, and w. We find that z is determined by the two smaller of the three PDF exponents, and in some cases by only the smallest. A PDF with decay exponent less than 3 has a divergent second moment, and thus is a Levy distribution. When the smallest decay exponent is between 3/2 and 3, the motion is superdiffusive (1<z<2). When the smallest exponent is between 1 and 3/2, the motion can be subdiffusive (z<1); this is in contrast with the previous results.