Galactic cosmic rays (GCRs) are energetic, charged particles coming from outside the solar system. These particles encounter an outward-moving turbulent solar wind with cyclic magnetic-field fluctuations when entering the heliosphere. This causes convection and diffusion in the heliosphere. The GCR counts detected by the ground-based neutron-monitor stations show intensity changes with a fluctuation of ∼ 11 years and are anti-correlated with the sunspot numbers with some time lags. GCRs experience various types of modulation from different solar activity features and are important components of space weather. The previous solar cycle, Cycle 23, has shown anomalous behavior with a prolonged deep minimum, which was characterized by a record-setting high Galactic cosmic-ray flux observed at Earth. Solar Cycle 24 started much later than expected and progressed sluggishly toward its maxima. In this paper, we study the heliospheric modulation and intermediate-term periodicities of GCRs during the ascending phase of Cycle 24. We utilize simultaneous solar, interplanetary plasma, magnetic field, and geomagnetic activity data including the tilt angle of the heliospheric current sheet, and we study their relation with GCRs. The wavelet power spectrum of GCRs exhibits the presence of a variety of prominent short- and mid-term periodicities including the well-known Rieger and quasi-biennial periodicities. Possible explanations of the observed results are discussed in the light of numerical models.