Using the Lukasiewicz 3-valued implication operator, the notion of an (${alpha},{eta}$)-intuitionistic fuzzy left (right) $h$-ideal of a hemiring is introduced, where ${alpha},{eta}{in}{{in},q,{in}{wedge}q,{in}{vee}q}$. We define intuitionistic fuzzy left (right) $h$-ideal with thresholds ($s,t$) of a hemiring R and investigate their various properties. We characterize intuitionistic fuzzy left (right) $h$-ideal with thresholds ($s,t$) and (${alpha},{eta}$)-intuitionistic fuzzy left (right) $h$-ideal of a hemiring R by its level sets. We establish that an intuitionistic fuzzy set A of a hemiring R is a (${in},{in}$) (or (${in},{in}{vee}q$) or (${in}{wedge}q,{in}$)-intuitionistic fuzzy left (right) $h$-ideal of R if and only if A is an intuitionistic fuzzy left (right) $h$-ideal with thresholds (0, 1) (or (0, 0.5) or (0.5, 1)) of R respectively. It is also shown that A is a (${in},{in}$) (or (${in},{in}{vee}q$) or (${in}{wedge}q,{in}$))-intuitionistic fuzzy left (right) $h$-ideal if and only if for any $p{in}$ (0, 1] (or $p{in}$ (0, 0.5] or $p{in}$ (0.5, 1] ), $A_p$ is a fuzzy left (right) $h$-ideal. Finally, we prove that an intuitionistic fuzzy set A of a hemiring R is an intuitionistic fuzzy left (right) $h$-ideal with thresholds ($s,t$) of R if and only if for any $p{in}(s,t]$, the cut set $A_p$ is a fuzzy left (right) $h$-ideal of R.