  In this paper, we introduce and study new classes of Ap\'ery-type series
involving the multiple $t$-harmonic sums by combining the methods of iterated
integral and Fourier--Legendre series expansions, where the multiple
$t$-harmonic sums are a variation of multiple harmonic sums in which all the
summation indices are restricted to odd numbers only. Our approach also enables
us to generalize some old classes of Ap\'ery-type series involving harmonic
sums to those with products of harmonic sums and multiple $t$-harmonic sums. In
all of these series, the central binomial coefficients appear as $a_n^{\pm 1}$
or $a_n^{\pm 2}$ where $a_n=\binom{2n}{n}/4^n$. We show that every such series
can be expressed as either the real or the imaginary part of a $\mathbb
Q$-linear combination of colored multiple zeta values of level 4.
